Reconstructing a piece of scenery with polynomially many observations

Benjamini asked whether the scenery reconstruction problem can be solved using only polynomially many observations. In this article, we answer his question in the affirmative for an i.i.d. uniformly colored scenery on Z observed along a random walk path with bounded jumps. We assume the random walk is recurrent, can reach every integer with positive probability, and the number of possible single steps for the random walk exceeds the number of colors. For infinitely many l, we prove that a finite piece of scenery of length l around the origin can be reconstructed up to reflection and a small translation from the first p(l) observations with high probability; here p is a polynomial and the probability that the reconstruction succeeds converges to 1 as l→∞.